Question

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.f(x) = $$x^2$$−25 / x−5; a=5

Accepted Answer

Step 1: Identify the function and the point of interest. The function given is $$ f(x) = \frac{x^2 - 25}{x - 5} $$ and the point of interest is $$ a = 5 . $$Step 2: Simplify the function if possible. Notice that $$ x^2 - 25 $$ can be factored as $$ (x - 5)(x + 5) . $$Therefore, the function can be rewritten as $$ f(x) = \frac{(x - 5)(x + 5)}{x - 5} . $$Step 3: Analyze the simplified function. The $$ x - 5 $$ terms cancel out, leaving $$ f(x) = x + 5 $$ for $$ x 
eq 5 . $$This indicates that the function is a line with a hole at $$ x = 5 . $$Step 4: Determine the behavior of the function around $$ x = 5 . $$Since $$ f(x) = x + 5 $$ for $$ x 
eq 5 $$, as $$ x 	o 5^- $$ and $$ x 	o 5^+ $$, $$ f(x) 	o 10 . $$Thus, $$ \lim_{x 	o 5^-} f(x) = 10 $$ and $$ \lim_{x 	o 5^+} f(x) = 10 . $$Step 5: Conclude about the limits and the value of $$ f(a) . $$Since both one-sided limits are equal, $$ \lim_{x 	o 5} f(x) = 10 . $$However, $$ f(5)  is $$undefined because the original function has a zero in the denominator at $$ x = 5 .$$

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.
f(x) = x^2−25 / x−5; a=5

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Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.f(x) = x^2−25 / x−5; a=5